// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
// Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_JACOBI_H
#define EIGEN_JACOBI_H

namespace Eigen {

/** \ingroup Jacobi_Module
  * \jacobi_module
  * \class JacobiRotation
  * \brief Rotation given by a cosine-sine pair.
  *
  * This class represents a Jacobi or Givens rotation.
  * This is a 2D rotation in the plane \c J of angle \f$ \theta \f$ defined by
  * its cosine \c c and sine \c s as follow:
  * \f$ J = \left ( \begin{array}{cc} c & \overline s \\ -s  & \overline c \end{array} \right ) \f$
  *
  * You can apply the respective counter-clockwise rotation to a column vector \c v by
  * applying its adjoint on the left: \f$ v = J^* v \f$ that translates to the following Eigen code:
  * \code
  * v.applyOnTheLeft(J.adjoint());
  * \endcode
  *
  * \sa MatrixBase::applyOnTheLeft(), MatrixBase::applyOnTheRight()
  */
template <typename Scalar> class JacobiRotation
{
public:
    typedef typename NumTraits<Scalar>::Real RealScalar;

    /** Default constructor without any initialization. */
    EIGEN_DEVICE_FUNC
    JacobiRotation() {}

    /** Construct a planar rotation from a cosine-sine pair (\a c, \c s). */
    EIGEN_DEVICE_FUNC
    JacobiRotation(const Scalar& c, const Scalar& s) : m_c(c), m_s(s) {}

    EIGEN_DEVICE_FUNC Scalar& c() { return m_c; }
    EIGEN_DEVICE_FUNC Scalar c() const { return m_c; }
    EIGEN_DEVICE_FUNC Scalar& s() { return m_s; }
    EIGEN_DEVICE_FUNC Scalar s() const { return m_s; }

    /** Concatenates two planar rotation */
    EIGEN_DEVICE_FUNC
    JacobiRotation operator*(const JacobiRotation& other)
    {
        using numext::conj;
        return JacobiRotation(m_c * other.m_c - conj(m_s) * other.m_s, conj(m_c * conj(other.m_s) + conj(m_s) * conj(other.m_c)));
    }

    /** Returns the transposed transformation */
    EIGEN_DEVICE_FUNC
    JacobiRotation transpose() const
    {
        using numext::conj;
        return JacobiRotation(m_c, -conj(m_s));
    }

    /** Returns the adjoint transformation */
    EIGEN_DEVICE_FUNC
    JacobiRotation adjoint() const
    {
        using numext::conj;
        return JacobiRotation(conj(m_c), -m_s);
    }

    template <typename Derived> EIGEN_DEVICE_FUNC bool makeJacobi(const MatrixBase<Derived>&, Index p, Index q);
    EIGEN_DEVICE_FUNC
    bool makeJacobi(const RealScalar& x, const Scalar& y, const RealScalar& z);

    EIGEN_DEVICE_FUNC
    void makeGivens(const Scalar& p, const Scalar& q, Scalar* r = 0);

protected:
    EIGEN_DEVICE_FUNC
    void makeGivens(const Scalar& p, const Scalar& q, Scalar* r, internal::true_type);
    EIGEN_DEVICE_FUNC
    void makeGivens(const Scalar& p, const Scalar& q, Scalar* r, internal::false_type);

    Scalar m_c, m_s;
};

/** Makes \c *this as a Jacobi rotation \a J such that applying \a J on both the right and left sides of the selfadjoint 2x2 matrix
  * \f$ B = \left ( \begin{array}{cc} x & y \\ \overline y & z \end{array} \right )\f$ yields a diagonal matrix \f$ A = J^* B J \f$
  *
  * \sa MatrixBase::makeJacobi(const MatrixBase<Derived>&, Index, Index), MatrixBase::applyOnTheLeft(), MatrixBase::applyOnTheRight()
  */
template <typename Scalar> EIGEN_DEVICE_FUNC bool JacobiRotation<Scalar>::makeJacobi(const RealScalar& x, const Scalar& y, const RealScalar& z)
{
    using std::abs;
    using std::sqrt;

    RealScalar deno = RealScalar(2) * abs(y);
    if (deno < (std::numeric_limits<RealScalar>::min)())
    {
        m_c = Scalar(1);
        m_s = Scalar(0);
        return false;
    }
    else
    {
        RealScalar tau = (x - z) / deno;
        RealScalar w = sqrt(numext::abs2(tau) + RealScalar(1));
        RealScalar t;
        if (tau > RealScalar(0))
        {
            t = RealScalar(1) / (tau + w);
        }
        else
        {
            t = RealScalar(1) / (tau - w);
        }
        RealScalar sign_t = t > RealScalar(0) ? RealScalar(1) : RealScalar(-1);
        RealScalar n = RealScalar(1) / sqrt(numext::abs2(t) + RealScalar(1));
        m_s = -sign_t * (numext::conj(y) / abs(y)) * abs(t) * n;
        m_c = n;
        return true;
    }
}

/** Makes \c *this as a Jacobi rotation \c J such that applying \a J on both the right and left sides of the 2x2 selfadjoint matrix
  * \f$ B = \left ( \begin{array}{cc} \text{this}_{pp} & \text{this}_{pq} \\ (\text{this}_{pq})^* & \text{this}_{qq} \end{array} \right )\f$ yields
  * a diagonal matrix \f$ A = J^* B J \f$
  *
  * Example: \include Jacobi_makeJacobi.cpp
  * Output: \verbinclude Jacobi_makeJacobi.out
  *
  * \sa JacobiRotation::makeJacobi(RealScalar, Scalar, RealScalar), MatrixBase::applyOnTheLeft(), MatrixBase::applyOnTheRight()
  */
template <typename Scalar>
template <typename Derived>
EIGEN_DEVICE_FUNC inline bool JacobiRotation<Scalar>::makeJacobi(const MatrixBase<Derived>& m, Index p, Index q)
{
    return makeJacobi(numext::real(m.coeff(p, p)), m.coeff(p, q), numext::real(m.coeff(q, q)));
}

/** Makes \c *this as a Givens rotation \c G such that applying \f$ G^* \f$ to the left of the vector
  * \f$ V = \left ( \begin{array}{c} p \\ q \end{array} \right )\f$ yields:
  * \f$ G^* V = \left ( \begin{array}{c} r \\ 0 \end{array} \right )\f$.
  *
  * The value of \a r is returned if \a r is not null (the default is null).
  * Also note that G is built such that the cosine is always real.
  *
  * Example: \include Jacobi_makeGivens.cpp
  * Output: \verbinclude Jacobi_makeGivens.out
  *
  * This function implements the continuous Givens rotation generation algorithm
  * found in Anderson (2000), Discontinuous Plane Rotations and the Symmetric Eigenvalue Problem.
  * LAPACK Working Note 150, University of Tennessee, UT-CS-00-454, December 4, 2000.
  *
  * \sa MatrixBase::applyOnTheLeft(), MatrixBase::applyOnTheRight()
  */
template <typename Scalar> EIGEN_DEVICE_FUNC void JacobiRotation<Scalar>::makeGivens(const Scalar& p, const Scalar& q, Scalar* r)
{
    makeGivens(p, q, r, typename internal::conditional<NumTraits<Scalar>::IsComplex, internal::true_type, internal::false_type>::type());
}

// specialization for complexes
template <typename Scalar> EIGEN_DEVICE_FUNC void JacobiRotation<Scalar>::makeGivens(const Scalar& p, const Scalar& q, Scalar* r, internal::true_type)
{
    using numext::conj;
    using std::abs;
    using std::sqrt;

    if (q == Scalar(0))
    {
        m_c = numext::real(p) < 0 ? Scalar(-1) : Scalar(1);
        m_s = 0;
        if (r)
            *r = m_c * p;
    }
    else if (p == Scalar(0))
    {
        m_c = 0;
        m_s = -q / abs(q);
        if (r)
            *r = abs(q);
    }
    else
    {
        RealScalar p1 = numext::norm1(p);
        RealScalar q1 = numext::norm1(q);
        if (p1 >= q1)
        {
            Scalar ps = p / p1;
            RealScalar p2 = numext::abs2(ps);
            Scalar qs = q / p1;
            RealScalar q2 = numext::abs2(qs);

            RealScalar u = sqrt(RealScalar(1) + q2 / p2);
            if (numext::real(p) < RealScalar(0))
                u = -u;

            m_c = Scalar(1) / u;
            m_s = -qs * conj(ps) * (m_c / p2);
            if (r)
                *r = p * u;
        }
        else
        {
            Scalar ps = p / q1;
            RealScalar p2 = numext::abs2(ps);
            Scalar qs = q / q1;
            RealScalar q2 = numext::abs2(qs);

            RealScalar u = q1 * sqrt(p2 + q2);
            if (numext::real(p) < RealScalar(0))
                u = -u;

            p1 = abs(p);
            ps = p / p1;
            m_c = p1 / u;
            m_s = -conj(ps) * (q / u);
            if (r)
                *r = ps * u;
        }
    }
}

// specialization for reals
template <typename Scalar> EIGEN_DEVICE_FUNC void JacobiRotation<Scalar>::makeGivens(const Scalar& p, const Scalar& q, Scalar* r, internal::false_type)
{
    using std::abs;
    using std::sqrt;
    if (q == Scalar(0))
    {
        m_c = p < Scalar(0) ? Scalar(-1) : Scalar(1);
        m_s = Scalar(0);
        if (r)
            *r = abs(p);
    }
    else if (p == Scalar(0))
    {
        m_c = Scalar(0);
        m_s = q < Scalar(0) ? Scalar(1) : Scalar(-1);
        if (r)
            *r = abs(q);
    }
    else if (abs(p) > abs(q))
    {
        Scalar t = q / p;
        Scalar u = sqrt(Scalar(1) + numext::abs2(t));
        if (p < Scalar(0))
            u = -u;
        m_c = Scalar(1) / u;
        m_s = -t * m_c;
        if (r)
            *r = p * u;
    }
    else
    {
        Scalar t = p / q;
        Scalar u = sqrt(Scalar(1) + numext::abs2(t));
        if (q < Scalar(0))
            u = -u;
        m_s = -Scalar(1) / u;
        m_c = -t * m_s;
        if (r)
            *r = q * u;
    }
}

/****************************************************************************************
*   Implementation of MatrixBase methods
****************************************************************************************/

namespace internal {
    /** \jacobi_module
  * Applies the clock wise 2D rotation \a j to the set of 2D vectors of coordinates \a x and \a y:
  * \f$ \left ( \begin{array}{cc} x \\ y \end{array} \right )  =  J \left ( \begin{array}{cc} x \\ y \end{array} \right ) \f$
  *
  * \sa MatrixBase::applyOnTheLeft(), MatrixBase::applyOnTheRight()
  */
    template <typename VectorX, typename VectorY, typename OtherScalar>
    EIGEN_DEVICE_FUNC void apply_rotation_in_the_plane(DenseBase<VectorX>& xpr_x, DenseBase<VectorY>& xpr_y, const JacobiRotation<OtherScalar>& j);
}  // namespace internal

/** \jacobi_module
  * Applies the rotation in the plane \a j to the rows \a p and \a q of \c *this, i.e., it computes B = J * B,
  * with \f$ B = \left ( \begin{array}{cc} \text{*this.row}(p) \\ \text{*this.row}(q) \end{array} \right ) \f$.
  *
  * \sa class JacobiRotation, MatrixBase::applyOnTheRight(), internal::apply_rotation_in_the_plane()
  */
template <typename Derived>
template <typename OtherScalar>
EIGEN_DEVICE_FUNC inline void MatrixBase<Derived>::applyOnTheLeft(Index p, Index q, const JacobiRotation<OtherScalar>& j)
{
    RowXpr x(this->row(p));
    RowXpr y(this->row(q));
    internal::apply_rotation_in_the_plane(x, y, j);
}

/** \ingroup Jacobi_Module
  * Applies the rotation in the plane \a j to the columns \a p and \a q of \c *this, i.e., it computes B = B * J
  * with \f$ B = \left ( \begin{array}{cc} \text{*this.col}(p) & \text{*this.col}(q) \end{array} \right ) \f$.
  *
  * \sa class JacobiRotation, MatrixBase::applyOnTheLeft(), internal::apply_rotation_in_the_plane()
  */
template <typename Derived>
template <typename OtherScalar>
EIGEN_DEVICE_FUNC inline void MatrixBase<Derived>::applyOnTheRight(Index p, Index q, const JacobiRotation<OtherScalar>& j)
{
    ColXpr x(this->col(p));
    ColXpr y(this->col(q));
    internal::apply_rotation_in_the_plane(x, y, j.transpose());
}

namespace internal {

    template <typename Scalar, typename OtherScalar, int SizeAtCompileTime, int MinAlignment, bool Vectorizable> struct apply_rotation_in_the_plane_selector
    {
        static EIGEN_DEVICE_FUNC inline void run(Scalar* x, Index incrx, Scalar* y, Index incry, Index size, OtherScalar c, OtherScalar s)
        {
            for (Index i = 0; i < size; ++i)
            {
                Scalar xi = *x;
                Scalar yi = *y;
                *x = c * xi + numext::conj(s) * yi;
                *y = -s * xi + numext::conj(c) * yi;
                x += incrx;
                y += incry;
            }
        }
    };

    template <typename Scalar, typename OtherScalar, int SizeAtCompileTime, int MinAlignment>
    struct apply_rotation_in_the_plane_selector<Scalar, OtherScalar, SizeAtCompileTime, MinAlignment, true /* vectorizable */>
    {
        static inline void run(Scalar* x, Index incrx, Scalar* y, Index incry, Index size, OtherScalar c, OtherScalar s)
        {
            enum
            {
                PacketSize = packet_traits<Scalar>::size,
                OtherPacketSize = packet_traits<OtherScalar>::size
            };
            typedef typename packet_traits<Scalar>::type Packet;
            typedef typename packet_traits<OtherScalar>::type OtherPacket;

            /*** dynamic-size vectorized paths ***/
            if (SizeAtCompileTime == Dynamic && ((incrx == 1 && incry == 1) || PacketSize == 1))
            {
                // both vectors are sequentially stored in memory => vectorization
                enum
                {
                    Peeling = 2
                };

                Index alignedStart = internal::first_default_aligned(y, size);
                Index alignedEnd = alignedStart + ((size - alignedStart) / PacketSize) * PacketSize;

                const OtherPacket pc = pset1<OtherPacket>(c);
                const OtherPacket ps = pset1<OtherPacket>(s);
                conj_helper<OtherPacket, Packet, NumTraits<OtherScalar>::IsComplex, false> pcj;
                conj_helper<OtherPacket, Packet, false, false> pm;

                for (Index i = 0; i < alignedStart; ++i)
                {
                    Scalar xi = x[i];
                    Scalar yi = y[i];
                    x[i] = c * xi + numext::conj(s) * yi;
                    y[i] = -s * xi + numext::conj(c) * yi;
                }

                Scalar* EIGEN_RESTRICT px = x + alignedStart;
                Scalar* EIGEN_RESTRICT py = y + alignedStart;

                if (internal::first_default_aligned(x, size) == alignedStart)
                {
                    for (Index i = alignedStart; i < alignedEnd; i += PacketSize)
                    {
                        Packet xi = pload<Packet>(px);
                        Packet yi = pload<Packet>(py);
                        pstore(px, padd(pm.pmul(pc, xi), pcj.pmul(ps, yi)));
                        pstore(py, psub(pcj.pmul(pc, yi), pm.pmul(ps, xi)));
                        px += PacketSize;
                        py += PacketSize;
                    }
                }
                else
                {
                    Index peelingEnd = alignedStart + ((size - alignedStart) / (Peeling * PacketSize)) * (Peeling * PacketSize);
                    for (Index i = alignedStart; i < peelingEnd; i += Peeling * PacketSize)
                    {
                        Packet xi = ploadu<Packet>(px);
                        Packet xi1 = ploadu<Packet>(px + PacketSize);
                        Packet yi = pload<Packet>(py);
                        Packet yi1 = pload<Packet>(py + PacketSize);
                        pstoreu(px, padd(pm.pmul(pc, xi), pcj.pmul(ps, yi)));
                        pstoreu(px + PacketSize, padd(pm.pmul(pc, xi1), pcj.pmul(ps, yi1)));
                        pstore(py, psub(pcj.pmul(pc, yi), pm.pmul(ps, xi)));
                        pstore(py + PacketSize, psub(pcj.pmul(pc, yi1), pm.pmul(ps, xi1)));
                        px += Peeling * PacketSize;
                        py += Peeling * PacketSize;
                    }
                    if (alignedEnd != peelingEnd)
                    {
                        Packet xi = ploadu<Packet>(x + peelingEnd);
                        Packet yi = pload<Packet>(y + peelingEnd);
                        pstoreu(x + peelingEnd, padd(pm.pmul(pc, xi), pcj.pmul(ps, yi)));
                        pstore(y + peelingEnd, psub(pcj.pmul(pc, yi), pm.pmul(ps, xi)));
                    }
                }

                for (Index i = alignedEnd; i < size; ++i)
                {
                    Scalar xi = x[i];
                    Scalar yi = y[i];
                    x[i] = c * xi + numext::conj(s) * yi;
                    y[i] = -s * xi + numext::conj(c) * yi;
                }
            }

            /*** fixed-size vectorized path ***/
            else if (SizeAtCompileTime != Dynamic && MinAlignment > 0)  // FIXME should be compared to the required alignment
            {
                const OtherPacket pc = pset1<OtherPacket>(c);
                const OtherPacket ps = pset1<OtherPacket>(s);
                conj_helper<OtherPacket, Packet, NumTraits<OtherPacket>::IsComplex, false> pcj;
                conj_helper<OtherPacket, Packet, false, false> pm;
                Scalar* EIGEN_RESTRICT px = x;
                Scalar* EIGEN_RESTRICT py = y;
                for (Index i = 0; i < size; i += PacketSize)
                {
                    Packet xi = pload<Packet>(px);
                    Packet yi = pload<Packet>(py);
                    pstore(px, padd(pm.pmul(pc, xi), pcj.pmul(ps, yi)));
                    pstore(py, psub(pcj.pmul(pc, yi), pm.pmul(ps, xi)));
                    px += PacketSize;
                    py += PacketSize;
                }
            }

            /*** non-vectorized path ***/
            else
            {
                apply_rotation_in_the_plane_selector<Scalar, OtherScalar, SizeAtCompileTime, MinAlignment, false>::run(x, incrx, y, incry, size, c, s);
            }
        }
    };

    template <typename VectorX, typename VectorY, typename OtherScalar>
    EIGEN_DEVICE_FUNC void /*EIGEN_DONT_INLINE*/
    apply_rotation_in_the_plane(DenseBase<VectorX>& xpr_x, DenseBase<VectorY>& xpr_y, const JacobiRotation<OtherScalar>& j)
    {
        typedef typename VectorX::Scalar Scalar;
        const bool Vectorizable =
            (int(VectorX::Flags) & int(VectorY::Flags) & PacketAccessBit) && (int(packet_traits<Scalar>::size) == int(packet_traits<OtherScalar>::size));

        eigen_assert(xpr_x.size() == xpr_y.size());
        Index size = xpr_x.size();
        Index incrx = xpr_x.derived().innerStride();
        Index incry = xpr_y.derived().innerStride();

        Scalar* EIGEN_RESTRICT x = &xpr_x.derived().coeffRef(0);
        Scalar* EIGEN_RESTRICT y = &xpr_y.derived().coeffRef(0);

        OtherScalar c = j.c();
        OtherScalar s = j.s();
        if (c == OtherScalar(1) && s == OtherScalar(0))
            return;

        apply_rotation_in_the_plane_selector<Scalar,
                                             OtherScalar,
                                             VectorX::SizeAtCompileTime,
                                             EIGEN_PLAIN_ENUM_MIN(evaluator<VectorX>::Alignment, evaluator<VectorY>::Alignment),
                                             Vectorizable>::run(x, incrx, y, incry, size, c, s);
    }

}  // end namespace internal

}  // end namespace Eigen

#endif  // EIGEN_JACOBI_H
